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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Similarities and differences between Bose and Fermi gases

Raj K. Pathria

Published Mar 1, 1998• 25 citations
Publisher

Abstract

The purported ``equivalence'' between noninteracting Bose and Fermi gases in two dimensions is critically examined. Staying away from the thermodynamic limit, an exact calculation reveals some subtle, but significant, differences between the properties of the two systems that arise characteristically from the statistics obeyed by them. At the same time, certain physical quantities show differences that are not so subtle, in that they remain present even when the thermodynamic limit is taken. The role played by the spatial dimensionality $d$ is also elucidated and it turns out that there is nothing fundamentally special about dimensionality 2; if the single-particle energy spectrum operating in the systems is of the form $\ensuremath{\varepsilon}\ensuremath{\propto}{p}^{s},$ then the equivalence in question arises when and only when $d=s.$

DOI
10.1103/physreve.57.2697

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The purported ``equivalence'' between noninteracting Bose and Fermi gases in two dimensions is critically examined. Staying away from the thermodynamic limit, an exact calculation reveals some subtle, but significant, differences between the properties of the two systems that arise characteristically from the statistics obeyed by them. At the same time, certain physical quantities show differences that are not so subtle, in that they remain present even when the thermodynamic limit is taken. The role played by the spatial dimensionality ddd is also elucidated and it turns out that there is nothing fundamentally special about dimensionality 2; if the single-particle energy spectrum operating in the systems is of the form \ensuremathε\ensuremath∝ps,\ensuremath{\varepsilon}\ensuremath{\propto}{p}^{s},\ensuremathε\ensuremath∝ps, then the equivalence in question arises when and only when d=s.d=s.d=s.
I discuss ideal and interacting quantum gases obeying general fractional exclusion statistics. For systems with constant density of single-particle states, described in the mean field approximation, the entropy depends neither on the microscopic exclusion statistics, nor on the interaction. Such systems are called thermodynamically equivalent and I show that the microscopic reason for this equivalence is a one-to-one correspondence between the excited states of these systems. This provides a method, different from the bosonization technique, of transforming between systems of different exclusion statistics. In the last section the macroscopic aspects of this method are discussed. In appendix A, I calculate the fluctuation of the ground-state population of a condensed Bose gas in a grand-canonical ensemble and mean field approximation, while in appendix B I show a situation where although the system exhibits fractional exclusion properties on microscopic energy intervals, a rigorous calculation of the population of single-particle states reveals a condensation phenomenon. This also implies a malfunction of the usual and simplified calculation technique of the most probable statistical distributions.
We consider a model for the relativistic Bose gases with certain forms of the energy spectra in ddd spatial dimensions. First we discuss how some of the effects of the interactions can be brought into the energy spectra for independent particles. The thermodynamical properties arising from the energy spectra are calculated with particular attention to the spatial dependence of Bose-Einstein condensation. Then we explicitly interpret some special cases of the energy spectra through Fourier transformation to the usual coordinate-dependent potentials.
We consider two-dimensional imperfect attractive Fermi and repulsive Bose gases consisting of spinless point particles whose total interparticle interaction energy is represented by aN2/2Va{N}^{2}/2VaN2/2V with $a=\ensuremath{-}{a}_{F}\ensuremath{\le}0$ for fermions and $a={a}_{B}\ensuremath{\ge}0$ for bosons. We show that, in spite of the attraction, the thermodynamics of a d=2d=2d=2 imperfect Fermi gas remains well defined for $0\ensuremath{\le}{a}_{F}\ensuremath{\le}{a}_{0}={h}^{2}/2\ensuremath{\pi}m$, and is exactly the same as the one of the repulsive imperfect Bose gas with ${a}_{B}={a}_{0}\ensuremath{-}{a}_{F}$. In particular, for aF=a0{a}_{F}={a}_{0}aF​=a0​ one observes the thermodynamic equivalence of the attractive imperfect Fermi gas and the ideal Bose gas.
By expanding the grand partition function, we obtain the expression for partition function of two dimensional free particles which is seen to be of the same form as that of one dimensional harmonic oscillators. For such systems it is shown that the difference in statistics does not appear in thermal behavior. When the energy levels of the particles are modified as ε n 1 , n 2 = Δ + b ( n 1 2 + n 2 2 ), Δ >0, the Bose condensation occurs and, if we set Δ =2 b , the condensation temperature qualitatively agrees with that of Osborne. The speciffic heat of free Bosons in a narrow box is calculated as the function of temperature, its width d being taken 7.1 Å and 14.2 Å. The results show that the maximum of the specific heat occurs at higher temperatures than that of the case d =∞, and its height becomes lower. The weak excitation of a Fermi system can be treated as excitons which obey Bose statistics. And we can show, by the method analogous to that of Ward and Wilks and Dingle, that the sound wave in a Fermi gas should be considered as the second sound, or the propagation of fluctuation in exciton distribution. Its propagation velocity is shown to become p0/3mp_{0}/\sqrt{3}mp0​/3​m, where p 0 is the momentum at the Fermi surface.
The number-theoretical problem of partition of an integer corresponds to D=2D=2D=2. This problem obeys the Bose--Eeinstein statistics, where repeated terms are admissible in the partition, and to the Fermi--Dirac statistics, where they are inadmissible. The Hougen--Watson P,Z-diagram shows that this problem splits into two cases: the positive pressure domain corresponds to the Fermi system, and the negative, to the Bose system. This analogy can be applied to the van der Waals thermodynamics. The thermodynamic approach is related to four potentials corresponding to the energy, free energy, thermodynamic Gibbs potential, enthalpy. The important notion of de Broglie's wavelength permits passing from particle to wave packet, in particular, to Bose and Fermi distributions. Particles of ideal Bose and Fermi gases and the decay of a boson consisting of two fermions into separate fermions are studied. The case of finitely many particles NNN of the order of 10210^2102 is considered by heuristic considerations like those Fock used to derive the Hartree--Fock equation. The case of N≪1N\ll1N≪1 is studied by Gentile statistics, tropical geometry and nonstandard analysis (Leibnitz differential or monad). A relation for the energy of neutron separation from the atomic nucleus is obtained when the atomic nucleus volume and de Broglie's wavelength are known. The Appendix is author's paper written in 1995.
Within an exact microcanonical (MC) ensemble, we study the nonanalyticities of thermodynamic functions research in finite noninteracting Bose gases in traps. The results show that there exists a rich oscillatory behavior of MC thermodynamical quantities as a function of a system's total energy E (e.g., nonmonotonous temperature, nonanalytic and negative specific heats, and microscopic phase transitions). The origin of these nonanalyticities comes directly from the inverted curvature entropy S(E) with respect to E and the behaviors are different in different trap geometries, boundary conditions, and energy spectrum configurations. Contrary to the usual grandcanonical and canonical results, there exists Bose condensation and the nonanalyticities in the two-dimensional finite noninteracting Bose systems with different traps. We also discuss the critical temperature dependence on the particle number N with different ensembles, traps, and boundary conditions. In large enough N, almost all the results of the thermodynamical quantities become smooth, which are similar to the usual canonical behaviors. We emphasize the finite-size effects on the MC entropy change, which should, in principle, be observable in suitably designed experiments of the small systems.